What is meant by the expression 'Hubble time'?

For $a(t)$ the scale factor of the universe and assuming $a(0)=0$ and $a\left(t_{0}\right)=1$, where $t_{0}$ is the time now, obtain a formula for the size of the particle horizon $R_{0}$ of the universe.

Taking

$$a(t)=\left(t / t_{0}\right)^{\alpha},$$

show that $R_{0}$ is finite for certain values of $\alpha$. What might be the physically relevant values of $\alpha$ ? Show that the age of the universe is less than the Hubble time for these values of $\alpha$.